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Topic: Comparing Compactifactions
Replies: 6   Last Post: Mar 20, 2013 10:30 PM

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: Comparing Compactifactions
Posted: Mar 20, 2013 10:07 AM
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On Tue, 19 Mar 2013 22:36:42 -0700, William Elliot <>

>On Mon, 18 Mar 2013, David C. Ullrich wrote:
>> >Let (f,X) and (y,Y) be compactifications of S.
>> >Assume h in C(Y,X) and f = hg.
>> >
>> >Thue h is a continuous surjection and when Y is Hausdorff
>> >a closed quotient map.
>> >
>> >k = h|g(S):g(S) -> f(S) is a continuous bijection.
>> >It it a homeomorphism? If so, what's a proof like?

>> I doubt that it follows that k is a homeomorphism,
>> although I'm not going to try to give a counterexample.

>I'll have to check this out, but isn't
>k = fg^-1:g(S) -> f(S) a homeomorphism?

That seems clear. Don't know what I was thinking, sorry.


>> Assuming it doesn't follow, then k being a homeomorphism
>> "should" be part of the _definition_ of the partial
>> order on the class of compactifications that we're
>> struggling to define here.

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